Find parametric equations of a line that intersect both L1 and L2 at right angles.
L1: [x,y,z]=[4,8,-1] + t[2,3,-4] L2: (x-7)/-6 = (y-2)/1 = (z+1)/2

Each line is parallel to a vector -- can you identity a vector to which L1 is parallel? and likewise for L2? (Hint: think about the "slope" vector).

Once you have two vectors, it should be easy to find a vector perpendicular to these two (cross product). Think of this new vector as the "slope" vector of the desired line. All that is left is to ensure that the desired line intersects L1 and L2.